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Fast Zeta Transforms for Lattices with Few Irreducibles [chapter]

Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto, Jesper Nederlof, Pekka Parviainen
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arithmetic circuits of size O(vn) for computing the zeta  ...  transform and its inverse, thus enabling fast multiplication in the Möbius algebra.  ...  The authors are grateful to the anonymous reviewers for their valuable comments that helped to considerably strengthen the paper.  ... 
doi:10.1137/1.9781611973099.113 dblp:conf/soda/BjorklundKHNKP12 fatcat:cbllrdfesbaxhgilu2qaoxpthe

Fast Zeta Transforms for Lattices with Few Irreducibles

Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto, Jesper Nederlof, Pekka Parviainen
2015 ACM Transactions on Algorithms  
We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arithmetic circuits of size O(vn) for computing the zeta  ...  transform and its inverse, thus enabling fast multiplication in the Möbius algebra.  ...  The authors are grateful to the anonymous reviewers for their valuable comments that helped to considerably strengthen the paper.  ... 
doi:10.1145/2629429 fatcat:6nyk7cqcvfa35hivwnakaf6day

Fast Möbius inversion in semimodular lattices and U-labelable posets [article]

Petteri Kaski, Jukka Kohonen, Thomas Westerbäck
2016 arXiv   pre-print
We consider the problem of fast zeta and Möbius transforms in finite posets, particularly in lattices.  ...  It has previously been shown that for a certain family of lattices, zeta and Möbius transforms can be computed in O(e) elementary arithmetic operations, where e denotes the size of the covering relation  ...  We thank Marcus Greferath for the conjecture that became Theorem 1, and Matti Karppa for helpful comments.  ... 
arXiv:1603.03889v1 fatcat:33qk3ht4tfhobbmw2l7cldeqze

Page 76 of Mathematical Reviews Vol. , Issue 93a [page]

1993 Mathematical Reviews  
Finally, two conjectures are stated which assert that, with a few listed exceptions, there is a primitive (or irreducible) monic polynomial over a finite field of arbitrary degree with a prescribed coefficient  ...  This is achieved with the help of the Euler- Poincaré characteristic of irreducible components of a resolution of f(x).  ... 

Efficient Möbius Transformations and their applications to Dempster-Shafer Theory: Clarification and implementation [article]

Maxime Chaveroche, Franck Davoine, Véronique Cherfaoui
2021 arXiv   pre-print
We show that the complexity of the EMT is always inferior to the complexity of algorithms that consider the whole lattice, such as the Fast M\"obius Transform (FMT) for all DST transformations.  ...  In this paper, we propose sequences of graphs for the computation of the zeta and M\"obius transformations that optimally exploit both the structure of distributive semilattices and the information contained  ...  As pointed out in [17] , a special ordering of the join-irreducible elements of a lattice when using the Fast Zeta Transform [15] leads to the optimal computation of its zeta and Möbius transforms.  ... 
arXiv:2107.07359v1 fatcat:vklq42egybdf7f3zhimczoniy4

Fourier inversion for finite inverse semigroups [article]

Martin E. Malandro
2013 arXiv   pre-print
Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as fast Fourier and inverse Fourier transforms for the planar  ...  This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups.  ...  [3] , which finds small circuits for computing zeta transforms and Möbius transforms of arbitrary functions on lattices with few join-irreducibles.  ... 
arXiv:1212.6462v2 fatcat:akf63gi2gbenbexzqt6ema2a4i

Strong spectral gaps for compact quotients of products of PSL(2,ℝ)

Dubi Kelmer, Peter Sarnak
2009 Journal of the European Mathematical Society (Print)  
This note is concerned with the spectral gap for an irreducible co-compact lattice in G = PSL(2, R) d for d ≥ 2, which is the simplest and most basic case where the congruence subgroup property is not  ...  For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg conjectures.  ...  Venkataramana for discussions about various aspects of the paper. The first author was partially supported by the NSF grant DMS-0635607, and the second author by the NSF grant DMS-0758299.  ... 
doi:10.4171/jems/151 fatcat:v67snqy5p5gonp722fg76n4usq

Strong Spectral Gaps for Compact Quotients of Products of (2,) [article]

Dubi Kelmer, Peter Sarnak
2009 arXiv   pre-print
This note is concerned with the strong spectral gap for an irreducible co-compact lattice Γ in G=(2,)^d for d≥ 2 which is the simplest and most basic case where the congruence subgroup property is not  ...  For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures.  ...  We thank A.Gamburd and T.Venkataramana for discussions about various aspects of the paper.  ... 
arXiv:0808.2368v2 fatcat:nci5arahxjdk5f3vbwgoqyxq3y

Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra

R. Vehkalahti, Hsiao-Feng Lu, L. Luzzi
2013 IEEE Transactions on Information Theory  
This work concentrates on the study of inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic space-time block codes.  ...  Gorodnik for answering questions concerning point counting in Lie groups and the reviewers for their hard work that has benefited us greatly.  ...  Let us shortly describe a few of them.  ... 
doi:10.1109/tit.2013.2266396 fatcat:usnadpzeafewvekyc227qve4iy

Rummukainen-Gottlieb formula on a two-particle system with different masses

Ziwen Fu
2012 Physical Review D  
The finite size formula are achieved for both C_4v and C_2v symmetries. Our analytical results will be very helpful for the study of some resonances, such as kappa, vector kaon, and so on.  ...  Lüscher established a non-perturbative formula to extract the elastic scattering phases from two-particle energy spectrum in a torus using lattice simulations.  ...  The author thank Naruhito Ishizuka for kindly helping us about group symmetry.  ... 
doi:10.1103/physrevd.85.014506 fatcat:as52y7iwxrandbyyu2odsy2irq

On the Perturbation of Self-Organized Urban Street Networks [article]

Jerome Benoit, Saif Eddin Jabari
2019 arXiv   pre-print
Ultimately, we argue that the strength of the external surprisal drift might be an indicator for the disengagement of the city-dwellers for their city.  ...  We obtain the statistics for slightly drifted self-organized urban street networks.  ...  and meet-irreducible elements of the involved road-junction Galois lattice.]  ... 
arXiv:1903.06016v2 fatcat:pzdgrtkhyra4hbunqxuwzhbq3m

On the perturbation of self-organized urban street networks

Jérôme G. M. Benoit, Saif Eddin G. Jabari
2019 Applied Network Science  
Ultimately, we argue that the strength of the external surprisal drift might be an indicator for the disengagement of the city-dwellers for their city.  ...  We obtain the statistics for slightly drifted self-organized urban street networks.  ...  These join-irreducibles generate the state space with set union ∪ as join-operator.  ... 
doi:10.1007/s41109-019-0153-0 fatcat:642xrskjfjglzofcfis2f5nuyy

Fractality, Self-Similarity and Complex Dimensions [article]

Michel L. Lapidus, Machiel van Frankenhuijsen
2004 arXiv   pre-print
fast.  ...  For pedagogical reasons, our goal in § §2-6 is to illustrate the mathematical theory of complex dimensions of self-similar strings by means of a few representative examples and results, along with a discussion  ... 
arXiv:math/0401156v1 fatcat:vsmiup7xsfbmnorbfvvxdmmc6y

Representation growth of linear groups

Michael Larsen, Alexander Lubotzky
2008 Journal of the European Mathematical Society (Print)  
Let Γ be a group and r n (Γ) the number of its n-dimensional irreducible complex representations. We define and study the associated representation zeta function Z Γ (s) = ∞ n=1 r n (Γ)n −s .  ...  We end with some observations and conjectures regarding the global abscissa.  ...  Then for any two irreducible lattices Γ 1 and Γ 2 in H, ρ(Γ 1 ) = ρ(Γ 2 ).  ... 
doi:10.4171/jems/113 fatcat:awigcl7z2jeh7bvkplgpqrbw6u

Artin's Conjecture, Turing's Method, and the Riemann Hypothesis

Andrew R. Booker
2006 Experimental Mathematics  
In order to apply it we develop a rigorous algorithm for computing general L-functions on the critical line via the Fast Fourier Transform.  ...  In addition, we discuss two methods for locating zeros of arbitrary L-functions. The first uses the explicit formula and techniques developed in [BS05] for computing with trace formulae.  ...  Thus, the main Fourier transform is of 2 20 points, which takes only a few minutes to compute.  ... 
doi:10.1080/10586458.2006.10128976 fatcat:2lskjrcidfbdrama3sycxumvoq
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